Find the width of the tower at a height of 137 feet.

Question

The sides of a nuclear power plant cooling tower form a hyperbola. The diameter of the bottom of the tower is 280 feet. The smallest diameter of the tower is 136 feet which is 424.5 feet above the ground. The tower is 553 feet tall.

Find the width of the tower at a height of 137 feet.

Answer 1

Hi Guadalupe!

For this problem we need to refer to the definition of a hyperbola, which is:

x²/a² - y²/b² = 1

In this equation, x represents the radius of the tower (or the distance from the y-axis, which would run up the center of the tower), but we can replace x with the diameter (D) of the tower by using this substitution:

x = D/2

In the hyperbola equation, y represents the distance from the thinnest point (x is smallest when y = 0), which is 424.5 feet above ground level, so we can replace y by the height (H) from the ground using this substitution:

y = H - 424.5

Using these substitutions in the hyperbola equation gives the equation:

(D/2)²/a² - (H-424.5)²/b² = 1

where a and b are scaling factors that allow us to use the other constraints given in the problem:

D = 280 when H = 0

D = 136 when H = 424.5

These constraints give use two equations:

(280/2)²/a² - (0-424.5)²/b² = 1

(136/2)²/a² - (424.5-424.5)²/b² = 1

The second equation simplifies:

68²/a² = 1

We can solve for a²:

a² = 68² = 4,624

We can then use this value in the first equation to solve for b²:

140²/68² - (-424.5)²/b² = 1

b² = 424.5²/((140/68)² - 1) ≈ 55,639

We can then develop the final equation:

(D/2)²/4624 - (H-424.5)²/55638 = 1

and solve for D in terms of H:

D = 2*[4624*(1 + (H-424.5)²/55638)]^1/2

Using this equation with H = 137 feet gives:

D = 2*[4624*(1 + (137-424.5)²/55638)]^1/2

D ≈ 214.4

The width of the tower at 137 feet above the ground is 214.4 feet.

I hope this helps!

Seth